Surface singularities appeared in the hyperbolic Schwarz map for the hypergeometric equation

Abstract

Surface singularities, swallowtail and cuspidal edge, appear in the hyperbolic Schwarz map for the hypergeometric differential equation. Such singularities are studied in detail. After an overview of classical staffs, the hypergeometric equation and the Schwarz map, the hyperbolic Schwarz map is introduced. We study the singularities of this map, whose target is the hyperbolic 3-space, and visualize its image when the monodromy group is a finite group or a typical Fuchsian group. Several confluences of swallowtails are also observed.